Denoting Indices of Tensors

I was wondering how the indices of tensors work. I do not understand how the indices of tensors in can be used. For example, \eta _{\mu \nu }, the metric tensor, is like a matrix, and x^{u} is a contravector. How does this extend to notations such as T{_{a}}^{bc} and T{_{ab}}^{c}?

The index notation is used in 2 ways. 1, it is used to denote the components of a tensor in some (arbitrary) coordinate system. And 2, it is often used to denote the tensor itself. The second way is a shortcut, but, strictly speaking, is an abuse of notation.

Lower indices indicate covariance while upper indices indicate contravariance. The tensors you wrote at the end are respectively once covariant twicce contravariant and twice covariant once contravariant.

Ok. Thanks for the tip about Latex. I was hoping to clarify that tensors are used in a certain context. Thank you for the information, I now understand how it can be used, particularly in the case of the Kronecker delta.